AP Calculus BC glossary

Taylor series

A Taylor series represents a function as an infinite polynomial whose coefficients come from the function's derivatives at a single centre point. Each coefficient is a derivative at the centre divided by the factorial of its order.

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

The construction forces the polynomial to match the function's value and every derivative at the centre, which is why it approximates so well nearby and why the factorials appear.

In practice you rarely differentiate repeatedly. Most series are built by substituting into, differentiating, or integrating a known series such as the ones for exe^x, sin⁡x\sin x, cos⁡x\cos x, and 11−x\frac{1}{1-x}.

Centred where

A Taylor series is always about a specific centre. Changing the centre changes every coefficient, so the centre must be stated.

Appears in: Unit 10: Infinite Sequences and Series (BC)